Prime group graded rings with applications to partial crossed products and Leavitt path algebras

TL;DR

This paper extends classical results on primeness in group graded rings to nearly epsilon-strongly graded rings, providing new characterizations for prime rings, partial skew group rings, crossed products, and Leavitt path algebras.

Contribution

It generalizes Passman's classical result to a broader class of rings and offers new characterizations for prime properties in various algebraic structures.

Findings

Characterization of prime nearly epsilon-strongly graded rings

Prime partial skew group rings and crossed products

Generalized prime Leavitt path algebra criteria

Abstract

In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime $s$-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over $s$-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime $s$-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.